Problem: Suppose $\displaystyle G$ is a group such that $\displaystyle g^2=e_G\quad\forall g\in G$. Prove that $\displaystyle G$ is abelian.
Proof (1): Note that $\displaystyle g^2=e\implies g=g^{-1}$. Therefore $\displaystyle \left(ab\right)=\left(ab\right)^{-1}=b^{-1}a^{-1}=ba$.
Proof (2): Using the above we can see that $\displaystyle ab=a\left(e_G\right)b=a(ab)^2b=(aa)ba(bb)=a^2bab^2 =ba$
Part b doesn't make sense. Is there a typo?